A090780 a(n) = n*Product_{p prime, p|n} (p - 1)/2.

1, 1, 3, 2, 10, 3, 21, 4, 9, 10, 55, 6, 78, 21, 30, 8, 136, 9, 171, 20, 63, 55, 253, 12, 50, 78, 27, 42, 406, 30, 465, 16, 165, 136, 210, 18, 666, 171, 234, 40, 820, 63, 903, 110, 90, 253, 1081, 24, 147, 50, 408, 156, 1378, 27, 550, 84, 513, 406, 1711, 60, 1830, 465, 189

Offset: 1

Author

Benoit Cloitre, Feb 12 2004

Keywords

• nonn
• mult

Comments

a(2n+1) is the conjectured value of the length of period of sequence of Genocchi number of first kind read modulo (2n + 1) (cf. A001469).

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A023900, A034444. - R. J. Mathar, Feb 08 2011

Cf. A299822, A000010, A007947, A001221.

Programs

Maple

A023900 := proc(n) add( d*numtheory[mobius](d),d=numtheory[divisors](n)) ; end proc:
A001221 := proc(n) nops(numtheory[factorset](n)) ; end proc:
A076479 := proc(n) (-1)^A001221(n) ; end proc:
A034444 := proc(n) 2^A001221(n) ;end proc:
A090780 := proc(n) n/A076479(n)/A034444(n) *A023900(n); end proc:
seq(A090780(n),n=1..20) ; # R. J. Mathar, Apr 14 2011

Mathematica

a[n_] := Module[{f, p, e}, fun[p_, e_] := (p - 1)*p^e/2;
If[n == 1, 1, Times @@ (fun @@@ FactorInteger[n])]]; Array[a, 50] (* Amiram Eldar, Nov 23 2018 *)

PARI

a(n) = my(f=factor(n)[,1]); n*prod(k=1, #f, (f[k]-1)/2); \\ Michel Marcus, May 26 2019

PARI

a(n) = eulerphi(n)*factorback(factorint(n)[, 1]/2) \\ Jianing Song, Aug 11 2023

Formula

a(n) = (n/(-2)^omega(n))*(Sum_{d|n} d*mu(d)) = n*A023900(n)/(A076479(n)*A034444(n)).

a(n) = n*A173557(n)/2. - R. J. Mathar, Apr 14 2011

From Jianing Song, Nov 22 2018: (Start)

Multiplicative with a(p^e) = (p - 1)*p^e/2 = A000217(p-1)*p^(e-1).

a(n) = A299822(n)/2^A001221(n).

a(prime(n)) = A034953(n).

a(n) is odd if and only if n = A004614(k) or 2*A004614(k). (End)

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 2/(p-1)^2) = 5.72671092223951683002237367406848393189560038246828458038126468772919585... - Vaclav Kotesovec, Sep 20 2020

From Jianing Song, Aug 11 2023: (Start)

a(n) = phi(n) * Product_{p|n, p prime} (p/2), where phi = A000010.

Equals A000010(n)*A007947(n)/2^A001221(n). (End)